#72316800

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#342
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
socle	C₂²	C₂²×#342
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,10,11,16)(4,9)(5,14)(6,8)(7,12,15,13)(17,18,22,25)(19,26)(20,28,31,23)(21,24)(27,30)(29,32);(1,3,6,11)(2,7)(4,12,5,13)(8,15)(9,10)(14,16)(17,19,21,25)(18,22,24,26)(20,29,30,23)(27,32,28,31);(3,11)(7,15)(10,16)(12,13);(2,8)(7,15)(9,14)(10,16)(17,20,22,31)(18,23,25,28)(19,27,24,29)(21,30,26,32)

Polycyclic presentation:

2^10|a^2=g,b^2=efij,d^2=j,e^2=i,f^2=ij,b^a=be,c^b=cij,d^a=di,d^b=dj,e^a=eh,e^b=ehj,f^a=fij,f^b=fhj,f^e=fj,g^b=ghi,g^e=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316803, #72316852, #72317002
maximal normal subgroup of	#72316803, #72316852, #72317002

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5*F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);